Solutions to Alternating Current Circuits

Solutions to Alternating Current Circuits 1. A capacitor in an LC oscillator has a maximum potential difference of 15 V and a maximum energy of 360 J. At a certain instant the energy in the capacitor is 40 J. At that instant what is the potential difference across the capacitor? A) zero B) 5 V C) 10 V D) 15 V E) 20 V

1 Solution: U= C V 2 2

2U 2U 2 UV2 U 40mJ C = max V= =max = V = �15V 5 V V 2 max max C2 Umax U max 360mJ

Ans: B

2. In the circuit shown, switch S is first pushed up to charge the capacitor. When S is then pushed down, the current in the circuit will oscillate at a frequency of:

A) 318 Hz B) 0.01 Hz C) 12.500 Hz D) 2000 Hz

E) depends on V0

1 1 Solution: w = = = 2000rad / s LC (50创 10-3H)( 5 10 - 6 F ) w f= 318 Hz 2p

Ans: A

3. An LC series circuit with an inductance L and a capacitance C has an oscillation frequency f. Two inductors, each with inductance L, and two capacitors, each with capacitance C, are all wired in series and the circuit is completed. The oscillation frequency is: A) f/4 B) f/2 C) f D) 2f E) 4f

1 Solution: L= 2 L C= C eq eq 2

1 1 weq = = = w Leq C eq LC

Ans: C

4. An RLC circuit has a resistance of 200  and an inductance of 15 mH. Its oscillation frequency is 7000 Hz. At time t = 0 the current is 25 mA and there is no charge on the capacitor. After five complete cycles the current is: A) 0 B) 1.8  10–6 A C) 2.1  10–4 A D) 2.3  10–3 A E) 2.5  10–2 A

-Rt/2 L Solution: I( t) = I0 ecosw t

-(R/2 L) ( 5/ f ) I(5 cycle) = I0 e 轾 -3 200W 骣 5 =(25创 10A) exp 犏 - 琪 = 2.13 10-4 A 2 15 10-3 H 桫7000Hz 臌犏创( )

Ans: C

5. A circuit contains an inductor, a capacitor, and a light bulb connected as shown. In which frequency limit is the light bulb the brightest?

A) the low frequency limit B) the high frequency limit C) both

2 2 Solution: Pbulb= I bulb R bulb = I C R bulb

V V IC = IC = 2 1 2 骣1 R + R + 琪 iw C 桫wC

 IC & hence Pbulb are larger in the high frequency limit.

Ans: B

6. An RLC series circuit has L = 100 mH and C = 1 F. It is connected to a 1000-Hz source emf is found to lead the current by 75. The value of R is: A) 12.6  B) 126  C) 175  D) 1750  E) 1810  1 Solution: Z= R + iw L + iw C 1 wL - V= I Z = I Z ei f with Im Z w C tanf = = Re Z R 1 wL -  w C R = tanf 1 (2创 3.14 1000Hz)( 100� 10-3 H ) (2创 3.14 1000Hz)( 10-6 C) R = tan 75 籛125.7

Ans: B

7. In the diagram, the function y(t) = ymsin(t) is plotted as a solid curve. The other three curves have the form y(t) = ymsin(t + ), where  is between –/2 and +/2. Rank the curves according to the value of , from the most negative to the most positive.

A) 1, 2, 3 B) 2, 3, 1 C) 3, 2, 1 D) 1, 3, 2 E) 2, 1, 3

Solution: 1:  < 0 2:  > 0 3:  > 0, less than 2 Rank (- to +): 1,3,2 Ans: D

8. An RLC series circuit is driven by a sinusoidal emf with angular frequency d. If

d is increased without changing the amplitude of the emf the current amplitude increases. If the L is inductance, C is the capacitance, and R is the resistance, this means that:

A) d L > 1/d C

B) d L < 1/d C

C) d L = 1/d C

D) d L > R

E) d L < R

1 Solution: Z= R + iwd L + iwd C

V I =  |Z| must decrease when d increases. Z

2 2 骣 1 Z= R +琪wd L - 桫 wdC

d Z 1 1骣 1 骣 1 1 w L - < 0 =创 2琪wd L - 琪 L +2 < 0  d dwd2 Z桫 w d C 桫 w d C wdC

Ans: B

9. A series circuit consists of a 15- resistor, a 25-mH inductor, and a 35-F capacitor. If the frequency is 100 Hz the power factor is: A) 0 B) 0.20 C) 0.45 D) 0.05 E) 1.0

Solution: P= I V cosf

V= I Z  f= fZ 1 Z= R + iw L + iw C

R cosf = Z 2  2 骣 1 R+琪w L - 桫 w C

15 W = 2 2 轾 1 (15W) +犏( 2创 3.14 100Hz)( 25 � 10-3 H ) 2 3.14 100Hz 35 10-6 F 臌犏 ( 创 )( ) 0.45

Ans: C

10. A coil has a resistance of 60 and an impedance of 100. Its reactance, in ohms, is: A) 40 B) 60 C) 80 D) 117 E) 160

Solution: Z= R + iw L = R + i X L

2 2 2 22 2 2 Z= R + X L XL = Z - R =(100 W) -( 60 W) = 80 W

Ans: C